Let ${{u}_{n}}=\ln \left( \frac{n}{n+1} \right)+\frac{1}{n+1}-\frac{1}{n}$ , Find in terms of $N$, a

Question

Let ${{u}_{n}}=\ln \left( \frac{n}{n+1} \right)+\frac{1}{n+1}-\frac{1}{n}$ , Find in terms of $N$, an expression for ${{S}_{N}}$, where ${{S}_{N}}={{u}_{1}}+{{u}_{2}}+...+{{u}_{N}}$, simplifying your result as far as possible. Show that ${{S}_{N}}<0$ for all $N\ge 1$.

Accepted Answer

$S_N = -\ln(N+1) + \frac{1}{N+1} - 1$; $S_N < 0$

AJC P1 Q8 (a)

AJC P1 Q8 (a)

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